Lemma 72.9.5. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{ f_ i : X_ i \to X\} _{i \in I}$ is an fpqc covering of $X$. Then there exists a refinement $\mathcal{V} = \{ g_ i : T_ i \to X\}$ of $\mathcal{U}$ which is an fpqc covering such that each $T_ i$ is a scheme.

Proof. Omitted. Hint: For each $i$ choose a scheme $T_ i$ and a surjective étale morphism $T_ i \to X_ i$. Then check that $\{ T_ i \to X\}$ is an fpqc covering. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).